Inequalities Connecting the Annihilation and Independence Numbers

Abstract

Given a graph $G$, the number of its vertices is represented by $n(G)$, while the number of its edges is denoted as $m(G)$. An independent set in a graph is a set of vertices where no two vertices are adjacent to each other and the size of the maximum independent set is denoted by $\alpha(G)$. A matching in a graph refers to a set of edges where no two edges share a common vertex and the maximum matching size is denoted by $\mu(G)$. If $\alpha(G) + \mu(G) = n(G)$, then the graph $G$ is called a K\"{o}nig-Egerv\'{a}ry graph. Considering a graph $G$ with a degree sequence $d_1 \leq d_2 \leq \cdots \leq d_n$, the annihilation number $a(G)$ is defined as the largest integer $k$ such that the sum of the first $k$ degrees in the sequence is less than or equal to $m(G)$ (Pepper, 2004). It is a known fact that $\alpha(G)$ is less than or equal to $a(G)$ for any graph $G$. Our goal is to estimate the difference between these two parameters. Specifically, we prove a series of inequalities, including $a(G) - \alpha(G) \leq \frac{\mu(G) - 1}{2}$ for trees, $a(G) - \alpha(G) \leq 2 + \mu(G) - 2\sqrt{1 + \mu(G)}$ for bipartite graphs and $a(G) - \alpha(G) \leq \mu(G) - 2$ for K\"{o}nig-Egerv\'{a}ry graphs. Furthermore, we demonstrate that these inequalities serve as tight upper bounds for the difference between the annihilation and independence numbers, regardless of the assigned value for $\mu(G)$.